An integral
...s of the form $F+C$, where $C$ is a constant. Consequently, the indefinite integral \eqref{*} consists of all functions of the form $F+C$.
2 KB (292 words) - 20:53, 1 January 2019
A generalization of the narrow [[Denjoy integral|Denjoy integral]] introduced by A.Ya. Khinchin in [[#References|[1]]].
...ywhere. Sometimes the Khinchin integral is also called the Denjoy–Khinchin integral.
1 KB (157 words) - 08:39, 23 July 2014
...n to another variable of integration. For the [[Definite integral|definite integral]] the formula is
...alogue of \eqref{e:change_of_var} for the [[Indefinite integral|indefinite integral]] is the assertion that, if $F$ is a primitive of $f$, then $F\circ \phi$ i
2 KB (308 words) - 14:34, 17 November 2012
...real numbers, while $m$, $n$ and $p$ are rational numbers. The indefinite integral of a differential binomial,
...bers $p$, $(m+1)/n$ and $p+(m+1)/n$ is an integer. In all other cases, the integral of a differential binomial cannot be expressed by elementary functions (P.L
712 bytes (119 words) - 20:23, 1 January 2019
...is an additive set function. If $F$ is continuous, the indefinite Burkill integral is continuous as well.
...The Burkill integral is used in constructing the [[Denjoy integral|Denjoy integral]] in different spaces.
3 KB (563 words) - 09:54, 27 November 2018
The indefinite integral of the binomial differential
426 bytes (73 words) - 21:00, 9 December 2014
A method for isolating the algebraic part in indefinite integrals of rational functions. Let $ P( x) $
...ration of a rational fraction whose denominator has only simple roots; the integral of such a fraction is expressed through transcendental functions: logarithm
3 KB (482 words) - 15:56, 2 March 2022
...he derivative of an elementary function is also elementary; the indefinite integral of an elementary function cannot always be expressed in terms of elementary
1 KB (146 words) - 15:13, 4 May 2012
Finding the [[Strong derivative|strong derivative]] of an indefinite integral
is summable on $G$ (in particular, if $f\in L_p(G)$, $p>1$), then the integral $F$ of $f$ is strongly differentiable almost-everywhere on $G$. For any $\p
2 KB (271 words) - 14:20, 14 February 2020
...which are arbitrarily close to its indefinite [[Lebesgue integral|Lebesgue integral]]. The notion of majorant and minorant can be generalized to the case of ad
...<TR><TD valign="top">[2]</TD> <TD valign="top"> S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)</TD></TR></table>
2 KB (302 words) - 10:06, 15 April 2014
$#C+1 = 154 : ~/encyclopedia/old_files/data/I051/I.0501360 Integral calculus
...with it constitutes the foundation of mathematical analysis. The origin of integral calculus goes back to the early period of development of mathematics and it
22 KB (3,436 words) - 22:12, 5 June 2020
$#C+1 = 125 : ~/encyclopedia/old_files/data/M065/M.0605370 Multiple integral
...multiple integral (Riemann integral, Lebesgue integral, Lebesgue–Stieltjes integral, etc.).
11 KB (1,648 words) - 16:53, 20 January 2024
...lated forms of the integral constitutes the problem of [[Integral calculus|integral calculus]].
...ral science and technology, the notions of the indefinite and the definite integral have undergone a number of generalizations and modifications.
14 KB (2,170 words) - 17:21, 14 February 2020
The indefinite integral of sine is:
...interval $[-1,1]$ by $\phi(x)=\int_0^xdt/\sqrt{1-t^2}$. For $x=\pm1$ this integral is improper, but convergent. It is easy to see that $\phi$ is monotone incr
3 KB (441 words) - 13:51, 14 February 2020
The indefinite integral of the secant is
1 KB (202 words) - 15:13, 14 February 2020
$#C+1 = 18 : ~/encyclopedia/old_files/data/S110/S.1100170 Skorokhod integral
...nsion of the Itô stochastic integral (cf. [[Stochastic integral|Stochastic integral]]) introduced by A.V. Skorokhod in [[#References|[a8]]] in order to integra
4 KB (612 words) - 08:14, 6 June 2020
...on $(-\infty,\infty)$, then $f'(x)$ is $U$-almost-periodic; the indefinite integral $F(x)=\int_0^xf(t)dt$ is $U$-almost-periodic if $F(x)$ is a bounded functio
2 KB (376 words) - 10:22, 24 August 2014
The indefinite integral of the tangent is:
1 KB (237 words) - 14:16, 14 February 2020
...e fundamental differences in the reduction theory of positive-definite and indefinite quadratic forms.
==The reduction of indefinite quadratic forms.==
19 KB (2,708 words) - 08:06, 14 January 2024
...op">[a1]</TD> <TD valign="top"> M.G. Krein, "Topics in differential and integral equations and operator theory" , Birkhäuser (1983) (Translated from Russ
..."> I. [I. Gokhberg] Gohberg, P. Lancaster, L. Rodman, "Matrices and indefinite scalar products" , Birkhäuser (1983)</TD></TR>
2 KB (348 words) - 19:54, 26 November 2016
The term usually denotes a limiting process which yields a definition of integral of an unbounded function or of a function over an unbounded set, even when
...[a,b[$. If the limit exists and is finite, then one says that the improper integral converges and if not, that it diverges. A similar definition is possible fo
7 KB (1,076 words) - 12:30, 7 July 2013
is a polynomial, called the integral part of the fraction $ P / Q $,
in (1), can be found by the method of indefinite coefficients (cf. [[Undetermined coefficients, method of|Undetermined coeff
8 KB (1,257 words) - 03:49, 4 March 2022
...1 = 89 : ~/encyclopedia/old_files/data/K110/K.1100200 Kurzweil\ANDHenstock integral
...[[#References|[a10]]], gave an integral that is equivalent to the Lebesgue integral.
11 KB (1,718 words) - 22:15, 5 June 2020
and if the indefinite integral of $ f ( s) $
3 KB (422 words) - 19:35, 5 June 2020
are integers, the binary quadratic form is said to be integral. The expression $ d = ac - {b ^ {2} } /4 $
All integral automorphisms of a primitive form $ f $
7 KB (977 words) - 15:17, 31 March 2024
...ion, partial]]); the theory of integral equations (cf. [[Integral equation|Integral equation]]); [[Differential geometry|differential geometry]]; [[Variational
...es|Fourier series]], and Fourier integrals (cf. [[Fourier integral|Fourier integral]]).
19 KB (2,844 words) - 08:57, 13 January 2024
...by series (in particular, by trigonometric series), the primitive and the integral, relations between integration and differentiation, term-by-term integratio
...e integral|Lebesgue integral]]). These two fundamental ideas — measure and integral — formed the foundation of the metric theory of functions, which is conce
11 KB (1,633 words) - 08:00, 6 June 2020
a primitive of $u$, the integration by parts formula for the definite integral $\int_a^b
...The latter assertion is also called ''formula of integration by parts for indefinite integrals''.
3 KB (493 words) - 09:25, 16 August 2013
is called the indefinite inner product of the Krein space $ {\mathcal K} $.
then an indefinite inner product is defined on $ {\mathcal K} $
32 KB (4,628 words) - 10:55, 20 January 2024
...hinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin).
|valign="top"|{{Ref|Sa}}|| S. Saks, "Theory of the integral" , Hafner (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
5 KB (745 words) - 17:18, 18 August 2012
...uences of algebraic operations plus the operation of taking the indefinite integral (quadratures) of the functions given, then one speaks of the integration (i
4 KB (607 words) - 15:57, 14 February 2020
The indefinite integral $ F( x)= \int _ {0} ^ {x} f( t) dt $
5 KB (793 words) - 08:05, 6 June 2020
is an $ ( n + p ) $-dimensional indefinite [[Riemannian manifold|Riemannian manifold]] of index $ p $
...D></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S. Montiel, "An integral inequality for compact space-like hypersurfaces in a de Sitter space and ap
5 KB (698 words) - 13:01, 31 January 2022
(an analogue of the second metric of example 2)), and the integral metric
...ns which do not have all the properties 1)–3); such are, for example, an [[indefinite metric]], a [[symmetry on a set]], etc.
5 KB (793 words) - 09:16, 7 June 2016
...tions is larger than the number of equations; thus, they are also known as indefinite equations. In modern mathematics the concept of a Diophantine equation is a
...the scientists of Ancient Greece led to the study of rational solutions of indefinite equations. This point of view is systematically followed by Diophantus in h
18 KB (2,685 words) - 08:57, 18 August 2022
...and quotient representations, of decomposing representations into a direct integral, and of studying dual objects, and the problems of the theory of spherical
...an indefinite metric (cf. [[Space with an indefinite metric|Space with an indefinite metric]]). A complete description of the irreducible representations in suc
20 KB (2,775 words) - 16:40, 31 March 2020
itself either negative definite or indefinite, then the equilibrium is unstable.
...e to a system of equations with constant coefficients and have a quadratic integral of definite sign.
7 KB (1,104 words) - 13:23, 6 January 2024
...1 }$, which play an important role in the study of spectral properties of integral operators of Carleman type (cf. also [[Carleman operator|Carleman operator]
...d to both finite Hankel matrices, by considering discrete measures, and to indefinite Hankel matrices, by means of formal orthogonal polynomials. A survey of res
14 KB (2,126 words) - 16:45, 1 July 2020
...ed under a discrete change of the argument, as opposed to differential and integral calculus, where the argument changes continuously. Let $ f $
27 KB (3,799 words) - 17:14, 12 November 2023
...ightman functions based on the classical Lagrangian and the heuristic path integral (cf. also [[Quantum field theory|Quantum field theory]]).
...]]]; for models with $d$ arbitrary (however with a state space carrying an indefinite inner product), see e.g. [[#References|[a12]]].
9 KB (1,385 words) - 20:57, 8 February 2024
is called indefinite if it represents both positive and negative elements, and positive (negativ
...lving such equations reduces to the problem of representing integers by an integral quadratic form $ q $,
32 KB (4,863 words) - 08:08, 12 August 2022
...the infinite-dimensional case has led to the construction of a triangular integral representation. For completely-continuous operators $ B = B _ {R} + i B _
...nces|[20]]], [[#References|[21]]]), who obtained an expansion in a Fourier integral, connected with the non-self-adjoint problem
35 KB (5,059 words) - 04:12, 9 May 2022
...rion: A continuous function is convex if and only if it is the indefinite integral of a non-decreasing function.
9 KB (1,555 words) - 08:20, 25 July 2013
...s the cyclic vectors for the operator $S$, or function-theoretically by an integral representation formula arising from an absolutely continuous measure on the
...$\mathcal{M} = \theta H ^ { 2 }$ and $\theta ( e ^ { i t } )$ preserve an indefinite rather than Hilbert-space inner product) leads directly to the linear-fract
12 KB (1,802 words) - 17:01, 1 July 2020
...of A.L. Cauchy in the early 19th century. The creation of differential and integral calculus initiated a period of rapid development in mathematics and in rela
...st limit theorems (cf. [[Indefinite limits and expressions, evaluations of|Indefinite limits and expressions, evaluations of]]). Differential calculus is extensi
37 KB (5,252 words) - 11:45, 26 March 2023
...ange principle, and the principles of Gauss, Hertz, Chetaev, and Jourdain. Integral principles, which describe the properties of motion during any finite perio
in which the system will remain for an indefinite time if it was placed there with zero initial velocities $ {v _ \nu } (
29 KB (4,434 words) - 08:28, 6 June 2020
...function of $x$; then for any rational function $g(x, y)$, the indefinite integral
11 KB (1,916 words) - 00:44, 12 August 2019
...be replaced by a smaller number. The study of A.A. Markov on the minima of indefinite binary quadratic forms (cf. [[Binary quadratic form|Binary quadratic form]]
...ized and developed by C. Siegel, led him to the theorem that the number of integral points on algebraic curves of genus higher than zero is finite (cf. [[Dioph
54 KB (7,359 words) - 18:32, 31 March 2017
...generated by analytic operations on function spaces (such as differential, integral and difference operators) assumes a description of their spectra in terms o
...realized (in the simplest case of an operator with a cyclic vector) by the integral operator
16 KB (2,424 words) - 08:22, 6 June 2020
is bounded in integral norm, that is, $ \sup _ {t} \int _ {t} ^ {t+1} \| A ( \tau ) \| d \tau
or for which the integral $ \int \| B ( t) \| d t $
26 KB (3,943 words) - 20:22, 16 January 2024
lectures on integral invariants.
analogue of the indefinite integral, and a separator, which is an
12 KB (1,840 words) - 20:30, 6 March 2024
...analysis it includes differential and
integral transforms and the Fourier
integral.
...org/legacyimages/l/l059/l059340/l059340134.png" />; the linear operator of
indefinite integration, which takes <img align="absmiddle" border="0" src="https://www
67 KB (9,247 words) - 17:12, 29 October 2017
can be accessible; here the indefinite parameters $ \xi $
...re-evaluation during the process of the domain of admissible values of the indefinite parameters in the direction of their constraint. In this way, the problem o
32 KB (4,621 words) - 08:04, 6 June 2020
runs through all the integral points (i.e. points with integer coordinates) that satisfy some supplementa
of integral points and the distance function
28 KB (4,011 words) - 11:37, 19 January 2022
...al solutions of the equation $F_n(x,y)=a$, where $F_n(x,y)$ is a form with integral coefficients of degree $n\geq 3$ and $a$ is a non-zero integer, is finite (
where the integral is taken over the straight line $\Re s=\sigma>0$, $s=\sigma+it$. E.g., for
27 KB (4,516 words) - 18:38, 18 October 2014